Question: Simplify; express your answer in exponential form. Assume $q\neq 0, k\neq 0$. $\dfrac{{(q^{4}k^{5})^{-3}}}{{(q^{-4}k^{-5})^{4}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(q^{4}k^{5})^{-3} = (q^{4})^{-3}(k^{5})^{-3}}$ On the left, we have ${q^{4}}$ to the exponent ${-3}$ . Now ${4 \times -3 = -12}$ , so ${(q^{4})^{-3} = q^{-12}}$ Apply the ideas above to simplify the equation. $\dfrac{{(q^{4}k^{5})^{-3}}}{{(q^{-4}k^{-5})^{4}}} = \dfrac{{q^{-12}k^{-15}}}{{q^{-16}k^{-20}}}$ Break up the equation by variable and simplify. $\dfrac{{q^{-12}k^{-15}}}{{q^{-16}k^{-20}}} = \dfrac{{q^{-12}}}{{q^{-16}}} \cdot \dfrac{{k^{-15}}}{{k^{-20}}} = q^{{-12} - {(-16)}} \cdot k^{{-15} - {(-20)}} = q^{4}k^{5}$